Next, the stochastic optimization framework considered in [1]} is not directly applicable for UBSR optimization, as they assume that the objective function measurements have zero-mean noise, while UBSR estimation results in a noise component with a positive mean. The latter can be controlled using the batch size used for estimation. In [2]}, the authors derive non-asymptotic bound of the order \(O(\log n/\sqrt{n})\)  using a stochastic gradient algorithm for a biased stochastic optimization problem. However, their framework does not feature a batch size parameter, and their result require the existence of a Lyapunov function.
Finally, in [3]}, the author considers a biased gradient oracle, and provides a \(O(1/n)\)  bound. However, their results are not directly applicable for UBSR optimization, as they consider a deterministic bias parameter, and their result do not feature a tunable batch-size parameter.
